Splitting universal bundles over flag manifolds

Abstract:

Let ${\mathbf {F}}$ be one of the fields ${\mathbf {R}}$, ${\mathbf {C}}$, or ${\mathbf {H}}$ and correspondingly let ${\mathbf {F}}G$ be $O$, $U$, or ${\text {Sp}}$, i.e. the orthogonal, unitary, or symplectic group. Over the flag manifold ${\mathbf {F}}G({n_1} + \cdots + {n_k})/{\mathbf {F}}G({n_1}) \times \cdots \times {\mathbf {F}}G({n_k})$ one has vector bundles ${\gamma _i}$ over $F$ of dimension ${n_i}$, $1 \leqslant i \leqslant k$. This paper determines all cases in which ${\gamma _i}$ decomposes nontrivially as a Whitney sum.